In class we learned about point charges, electrostatic force, voltage, current etc. and discussed circuits along the way.
Now problems arise when I try to apply the learned concepts to explain how wires/circuits work:

I first thought about an electrical circuit as a 1-dimensional "restricted path" for charges to flow through. Voltage is then created by a surplus of charges (electrons) in one end and a lack of charges in the other one. However that brings up numerous problems that contradict stuff I read about circuits:

the electric field inside the wire would decrease with distance (inverse-square law), so if you measure voltage between two points along a circuit it would not be zero.

a non-uniform electric field would mean non-uniform current along the wire

when an ideal wire has no resistance, charges would keep accelerating, resulting in an ever-increasing current

So thinking about circuits as nothing but two connecting ends with different charge concentrations doesn't work. But then how do you explain electrical circuits with the basic concepts of electrodynamics?

I also read explanations that say that charges in a wire are not accelerated, but move uniformly. But if that is the case, then there should be no electrical field inside the wire or else the charges would accelerate. But without an electrical field, how can there be voltage, and what caused the charges to move through the wire in the first place?

It frustrated me for a long time now that I can't think of a "mechanical" (or should I say classical?) model of how charges actually move through a wire. Is there a cohesive model (for this supposedly simple problem) that arises from the basic concepts of electrodynamics?

$\begingroup$The mechanical analog for voltage is given by a mechanical potential. A potential is the amount of work something has to do to from a point A to a point B. In a potential field it is irrelevant on which path we are going from A to B. In case of an electric potential this quantity is proportional to the charge, so an electric potential is work divided by charge. A wire is simply a method to confine a charge to a certain path. In electrodynamics not all fields are potential fields and magnetic induction can not be described with a potential, though.$\endgroup$
– CuriousOneSep 6 '15 at 18:37

1 Answer
1

In class we learned about point charges, electrostatic force, voltage, current etc. and discussed circuits along the way.

And then you hopefully learned that voltage isn't a general concept, and that the scalar field is an entirely gauge dependent concept in electrodynamics.

I first thought about an electrical circuit as a 1-dimensional "restricted path" for charges to flow through.

If you aren't modeling the energy and momentum stored by the fields outside the wires then the physics is wrong. You might be able to get the right answers to some questions in some situations, but it is fundamentally wrong.

Voltage is then created by a surplus of charges (electrons) in one end and a lack of charges in the other one.

Nope. Let's start with the basics. There is this four dimensional vector called energy-momentum. If you pick a frame you might describe it in terms of $E,$ $p_x,$ $p_y,$ and $p_z.$ Electrons have momentum determined by their speed and mass. They have energy determined by their momentum and mass. The electromagnetic field has a momentum density determined by the electromagnetic field and can also store a nonzero amount of momentum on surfaces at infinity. The electromagnetic field has an energy density determined by the electromagnetic field and can also store a nonzero amount of energy on surfaces at infinity.

OK. That's real, the breakdown of the energy-momentum into energy and momentum is frame dependent, as is the splitting of the electromagnetic field into an electric and a magnetic field. But it's real. And energy and momentum are conserved. The field momentum isn't and the mechanical particle momentum isn't, but the total is conserved. Same with energy.

So let's talk about what makes things begin to move around. Forces. Forces make things begin to move around. We will tall about the Lorentz Force and other forces. The other forces will be things like you grabbing a circuit and moving it, or you pushing on two components to snap them together. We won't worry about the other forces, for our purposes they can exchange mechanical energy of other things with the mechanical energy of the charges. We will also treat the wire as basically having some nuclei and non mobile charges that collectively act reasonably like a solid positively charged object. So a big slab of uniform mass and (positive) charge density. That's just to try to be classical, not to be 100% accurate. But let's do the rest accurately.

So we want to talk about changes in energy and momentum. So let's go back to current. Recall how you can have current in a region with zero charge density. Similarly you have a flow of momentum and a flow of energy and the rate that they flow through a region isn't really directly related to the momentum density or the energy density at that point.

Which makes sense. An energy per cross sectional area let unit time has different units than an energy per unit volume so their numerical size is unrelated.

So in regions without current the field energy in a volume increases when there is a net flow in. And decreases when there is a net flow out. But when there is a current then work is done at a rate of $\vec J \cdot \vec E$ so when current is moving in the same direction as the electric field the charges gain mechanical energy (gain kinetic energy if no other forces) and the field loses an equal amount energy.

Similarly for momentum but now electric fields can exchange momentum even when there is no current as long as there is charge. And the rate at which the fields lose energy per volume is $\rho\vec E +\vec v\times\vec B$ which makes sense, the momentum the charges gain or lose is equal to the momentum the fields lose or gain at the same point and time.

But the energy or momentum of the fields at that point doesn't have to change. When there was no charge or current then the field momentum or field energy that flowed in had to increase the energy or momentum right there. But when charge or current is there then the field only changes the energy or momentum if the field energy flowing in minus the field energy given to the charges is nonzero. So far everything is reasonable.

the electric field inside the wire would decrease with distance (inverse-square law), so if you measure voltage between two points along a circuit it would not be zero.

This is a very bad way to think about it. Imagine you have a section of wire. Hold it horizontal and then start walking at a steady pace. Since we model the nuclei and nonmobile charges as a solid it will follow you. But what about the charges? It depends on the resistance of the material. If you pull the rod with you then for an instant one end moves and the electrons are left behind that leaves a charge imbalance on the two ends. Which can exert a force on the mobile charges to make them follow. But the electrons bump into the wire if it has resistance and lose mechanical energy.

The electrons feel a force due to the field. The field is directly equal to (assuming no incoming electromagnetic waves) exactly what Jefimenko's equations predict. Which is all determined by what the charge density looks like on the past light cone (so what it was like farther in the past when the charge is farther away, exactly far enough in the last that might just now had time to get from there to here, much like how we see stars based on what they were like back then). It is also determined by the time rate the charge density was changing on the past light cone and how the current density was changing in time back on the past light cone. (The magnetic field in contrast doesn't care about charge, just current density and the time rate of change of the current density.)

So eventually you can achieve a steady state where charge distributes itself basically all along the surface of the wire so that the mobile charges feel a force that does work at a rate equal to what it loses to heat the wire. Where did that energy come from? The mobile charges got it from the field but the nonmobile charge gave it to the field as they we dragged along by your hand. So ultimately pulling the wire acts like it has a bit more resistance to acceleration than just the mass of the nonmobile charges, because you have to overcome the force of the fields fighting the nonmobile you end up supplying the force you'd need to accelerate the mobile charges too, it's just that they move directly because of the fields.

Before we get to a detailed analysis of the walking wire let's look at what happens if you place a conducting wire of wire in a uniform constant electric field. The charge will arrange itself on the surface so that it produces an equal and opposite electric field in the wire so the total field is zero. Super two examples where charge arranges itself on the surface to provide uniform power to currents (though in the second example there is no current).

a non-uniform electric field would mean non-uniform current along the wire

In a stationary wire in say an RC circuit where the switch was thrown, the charge arranges itself so that there is a fairly uniform electric field inside the wire of the resistor. This allows the field to give the charges an equal amount of energy to what the charges give away to the wire as they heat it. Thus the can maintain a fairly constant velocity and a steady current throughout the resistor. So the charges arrange themselves on the surface so as to make a uniform field inside and that uniform field inside doesn't give the current more kinetic energy because they are losing just as much energy into heat.

when an ideal wire has no resistance, charges would keep accelerating, resulting in an ever-increasing current

An ideal wire without any resistance is a bit much to expect from a non quantum model. But recall that if you aren't going perfectly along the wire then you need to be going slowly enough that you don't fly out of the wire as you approach the edge. And you need some charge imbalances at the corner so that your steady flow gets the force it needs to make the corner. And so many assumptions about no going too fast are needed for the rather simple assumptions that you want the charges to stay inside the wires.

So thinking about circuits as nothing but two connecting ends with different charge concentrations doesn't work.

Right. Think of it as charge all along the surface. With even few mobile charges on the inside of corner and even more in the outside of corners and from straight segments have them arranged to have a nice uniform constant electric field inside to counter the resistance of the element.

You can arrange charge on the surface to make an electric field that countered a uniform external field so you arrange the charge to make a uniform field but this time it doesn't do it to oppose an external field it does it naturally.

Why? How? Let's look at how it started. You have an RC circuit and you start to close the switch. At first there is charge density all over the surface of one plate and the entire wire connected to it. And there is opposite charge density all over the other plate if the capacitor and all over the wire and resistor connected to it. But since the switch isn't thrown there is no current. And nothing changing.

So there is just an electric field and no magnetic field and the circuit is zero net charge so the electric field falls off fast enough spatially that no energy is stored st infinity. There is energy stored all through out space, every little bit of space except for inside the wires where there is no field right now.

Since this is statics there is a well defined potential and the outside surface of the wires are a constant potential surface. These surfaces are orthogonal to the electric field. It is important because energy flows orthogonally to the direction of the electric field. And energy also flows orthogonal to the magnetic field, so orthogonal to both fields.

So later we are going to imagine energy flowing orthogonally to the electric fields right now there are many equipotential surfaces layers between the two plates on the capacitor and layered between the two ends of the switch we haven't closed yet.

As you close the switch, the fields only change nearby at first. Effectively as the switch starts to close you have a large resistance. So a little bit of current needs a huge electric field. But all those equipotential surfaces got sandwiched between so there is a huge electric field right there so current can flow.

With the current come a magnetic field and with a magnetic field comes a floe of energy from the electric field energy. It comes in from the side right where those equipotential surfaces were when it was statics. And each bit of empty space starts to give its energy to the part next to it. You can imagine the energy flowing nearly along those equipotential surfaces from between the capacitor plates where most of the energy was stored, along the empty space outside the wires over to the circuit switch where it can give some kinetic energy to charges to start the current and to heat the wires. As the mobile charges actually transfer from one end to the other a current starts to developed all along the wire. When the electrons cross from the negative side to the positive they leave a charge imbalance behind that attracts more electrons from behind and so on. Similarly in the positively charge wires they already spread themselves (the holes the lack of enough electrons) out along the outside but the extra electrons coming in mean the already there electrons can scoot over. The changes in the electromagnetic field happen at the speed of light so the electrons everywhere really move because of the information of the other charges long before the charges get there themselves.

So whenever an electron moves it heats up the wire and slows down. So it needs a steady field to have a steady speed. So the electrons that got the information move away and the other ones don't follow as fast. This is like when we pulled the wire. The original change on the outside surface produced no field inside the change produces a field inside that helps it to move in a steady fashion. If the information cleared the way the charge imbalance makes positive charges to pull your electrons forward. If the slow down makes the charges behind you pile up the negative change imbalance pushes you forwards. Any deviation from the steady state creates the change imbalance that makes it approach the steady state. For instance when you go around a right turn the first charge on the left might not turn hard enough and then get farther out than normal and push others back into the wire and the next ones aren't flung out as much so charge stops piling up on the outside bank of the corner.

But then how do you explain electrical circuits with the basic concepts of electrodynamics?

So the charge on the outside arranges itself so the magnetic fields and electric fields outside transport the field energy through the space outside the wires to the places where the fields are. Where the resistor is the charges had to make stronger fields so it is like the equipotential surfaces got pushed along the wire until the current had to slow down to get across the resistor and then more surfaces for stuck in it. Really the energy was delivered to each electron as it heated the wire and sped up to its speed for the current. Coming in orthogonally to the electric field. And hence from outside the wire to the inside not along the wire.

But without an electrical field, how can there be voltage, and what caused the charges to move through the wire in the first place?

Inertia and exchanging momentum from the fields is what makes charges move. Voltage doesn't even make sense in electrodynamics.

The charges each react to field produced by past charges, consider studying the Jefimenko equations to see the fields and see the energy and momentum densities to know where and how the fields affect the particles.